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THE COMMUTING TRIPLES OF MATRICES
by
Yongho Han
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful¯llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2007
Copyright 2007 Yongho Han

The irreducibility of the variety C(m,n) commuting m-tuples of n X n matrices has been studied by several authors. It is known that this variety is irreducible if m=2, n=1,2,3 and reducible if m,n>=4. The irreducibility of C(3,n) was asked by Gersteinhaber. Until now we know that C(3,n) is reducible for n>=32, but irreducible for n<=4. -- If the characteristic of an arbitrary algebraically closed field is zero, then we have more specific answer for this question. In this case C(3,n) is reducible for n>=30 and irreducible for n<=6. -- This dissertation is primarily concerned with the study of the variety C(3,7). We show that this variety is also irreducible. Guralnick and Omladi\v{c} have conjectured that it is reducible for n > 7

THE COMMUTING TRIPLES OF MATRICES
by
Yongho Han
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful¯llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2007
Copyright 2007 Yongho Han